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Math Help - Continuous Functions!

  1. #1
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    Continuous Functions!

    Suppose h : (0,1)-> satisfies the following conditions:
    for all x Э (0,1) there exists d>0 s.t. for all x' Э (x, x+d)n(0,1) we have h(x)<=h(x')

    Prove that if h is continuous on (0,1) then h(x)<=h(y) whenever x,y Э (0,1) and x<=y. Use a counterexample to show that this results may not be true when h is continuous

    I thought I could do this by using definitions, but its not working for me.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by emjones View Post
    Suppose h : (0,1)-> satisfies the following conditions:
    for all x Э (0,1) there exists d>0 s.t. for all x' Э (x, x+d)n(0,1) we have h(x)<=h(x')

    Prove that if h is continuous on (0,1) then h(x)<=h(y) whenever x,y Э (0,1) and x<=y. Use a counterexample to show that this results may not be true when h is continuous

    I thought I could do this by using definitions, but its not working for me.
    I hate reading plain text

    Suppose 0,1)\to\mathbb{R}" alt="h0,1)\to\mathbb{R}" /> (presumably) satisfies the following conditions: for every x\in(0,1) there exists some \delta>0 such that x'\in (x,x+\delta)\implies h(x)\leqslant h(x'). Prove that if h is continuous then h is non-decreasing. Give a counter example to show the reverse is isn
    t true.
    Right? What have you tried?
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  3. #3
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    Hi drexel, sorry I don't know how to use the codes, but thank you for changing it for me.

    I have tried to use a corollary in my notes relating to monotonic functions and differentiability, where the MVT is applied to [x,y]<[a,b] but im not getting anywhere.

    Also I was asked to state the IVT earlier in the question, so thought this might be of use, but I don't think it can be used earlier! This is why im confused! Please help!
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by emjones View Post
    Hi drexel, sorry I don't know how to use the codes, but thank you for changing it for me.

    I have tried to use a corollary in my notes relating to monotonic functions and differentiability, where the MVT is applied to [x,y]<[a,b] but im not getting anywhere.

    Also I was asked to state the IVT earlier in the question, so thought this might be of use, but I don't think it can be used earlier! This is why im confused! Please help!
    You can't use the MVT! How do you know that f is differentiable? How do you think you'd use the IVT, that's what I'm thinking too.
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